48.3.9 problem Example 3.38

Internal problem ID [7533]
Book : THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section : Chapter 3. Ordinary Differential Equations. Section 3.5 HIGHER ORDER ODE. Page 181
Problem number : Example 3.38
Date solved : Sunday, March 30, 2025 at 12:13:52 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x x^{\prime \prime }-{x^{\prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 14

ode:=x(t)*diff(diff(x(t),t),t)-diff(x(t),t)^2 = 0; 
dsolve(ode,x(t), singsol=all);
 

\begin{align*} x &= 0 \\ x &= {\mathrm e}^{c_1 t} c_2 \\ \end{align*}

✓ Mathematica. Time used: 0.116 (sec). Leaf size: 14

ode=x[t]*D[x[t],{t,2}]-(D[x[t],t])^2==0; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to c_2 e^{c_1 t} \]

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(x(t)*Derivative(x(t), (t, 2)) - Derivative(x(t), t)**2,0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

NotImplementedError : The given ODE -sqrt(x(t)*Derivative(x(t), (t, 2))) + Derivative(x(t), t) cannot be solved by the factorable group method